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Integrable Hamiltonian systems

Course Code :2600WETIHS
Study domain:Mathematics
Academic year:2020-2021
Semester:1st semester
Contact hours:60
Credits:6
Study load (hours):168
Contract restrictions: No contract restriction
Language of instruction:English
Exam period:exam in the 1st semester
Lecturer(s)Sonja Hohloch

3. Course contents *

  1. Definition of Hamiltonian systems, basic properties & examples.
  2. Integrability in finite dimensions (Frobenius, Liouville, more general notions), basic properties, examples.
  3. Important examples of integrable systems in mathematics and physics: Spherical pendulum, rigid body, top (Lagrange, Euler, Kovalevskaya), coupled spin oscillators, coupled angular momenta...
  4. Local behaviour (regular points): Theorem of Arnold-Liouville, transformation to action-angle coordinates.
  5. Local behaviour (singular points): local normal form of Eliasson-Miranda-Zung for nondegenerate singular points in terms of hyperbolic, elliptic, and focus-focus components.
  6. Toric systems: Symplectic classification by Delzant by means of polygons.
  7. Semitoric systems: properties and interaction with the topology and geometry of the underlying manifold.
  8. Integrability in infinite dimensions ("integrable Hamiltonian partial differential equations"): motivation and important examples (Korteweg-de Vries, Sine-Gordon, Nonlinear Schrödinger equation).